On the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method

TL;DR

This paper develops easy-to-compute bounds on the entropy of sums of Bernoulli variables, using the Chen-Stein method, to assess the accuracy of Poisson approximations in information theory.

Contribution

It introduces novel bounds on the entropy of Bernoulli sums via the Chen-Stein method, linking information theory with Poisson approximation techniques.

Findings

Derived explicit bounds on entropy approximation error

Provided examples demonstrating bound applicability

Extended bounds to non-negative integer-valued variables

Abstract

This paper considers the entropy of the sum of (possibly dependent and non-identically distributed) Bernoulli random variables. Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived. The derivation of these bounds combines elements of information theory with the Chen-Stein method for Poisson approximation. The resulting bounds are easy to compute, and their applicability is exemplified. This conference paper presents in part the first half of the paper entitled "An information-theoretic perspective of the Poisson approximation via the Chen-Stein method" (see:arxiv:1206.6811). A generalization of the bounds that considers the accuracy of the Poisson approximation for the entropy of a sum of non-negative, integer-valued and bounded random variables is introduced in the full paper. It also derives lower bounds on the total variation distance, relative entropy and other measures that are not considered in this conference paper.